At my university, the Catholic campus ministry has mass three times each Sunday in the non-denominational campus chapel. The situation is a bit unusual for a couple reasons.
a) A group of students makes the bread used for communion (instead of the wafer hosts), so it does not keep as well and is broken into uneven pieces, so prolonged transportation risks crumbs.
b) There is no tabernacle in the chapel, but there is a (tiny) tabernacle off-campus at the small chapel in the campus ministry's house (it's really only big enough for a few small hosts and a monstrance). It is deemed not prudent to move the Eucharist there due to crumbs, the small size of the tabernacle, and the fact that the hosts are not paper-thin like the standard wafer hosts.
After communion, any remaining communion is brought to a back room of the non-denom chapel and (presumably reverently) consumed by extraordinary ministers since it is not moved off campus to the small tabernacle. I know EMHC's often finish any remaining Blood from the chalice at other masses I've been to, so I see nothing inherently wrong with this practice. But maybe you guys could give me some insight regarding whether EMHC's consuming the rest of the Body after communion is ok/reverent.